This looks quite similar like the "Lagrange models" defined by Joris van der Hoeven in https://hal.science/hal-01188378/document https://hal.science/hal-0118837

This looks quite similar like the "Lagrange models" defined by Joris van der Hoeven in https://hal.science/hal-01188378/document https://hal.science/hal-01188378/document, a version of Taylor models where the constant error term is replaced by an interval multiple of x^n. Representing functions locally is a very interesting problem: there are lots of possible approaches using intervals, Taylor series, Chebyshev series, etc. Big open design space if you ask me.